# Largest eigenvalues of a (random) correlation matrix?

I am recently studying on eigenvalues of a (random) correltion matrix. For a $$N\times N$$ correlation matrix (with a given meaning of randomness), its (1st, 2nd, etc.) eigenvalues have some distributions depending on $$N$$. For the case $$N=2$$, a correlation matrix is like $$\begin{bmatrix}1&x\\x&1\end{bmatrix}$$ with $$x\in[-1,1]$$ and its eigenvalues are $$1\pm x$$ and so its largest eigenvalue has an upper bound $$2$$. Also one can see that the distribution of the largest eigenvalue is actually a uniform distribution on $$[1,2]$$, when this correlation matrix is sampled uniformly ($$x$$ uniform in $$[-1,1]$$).

One would be interested to know the analogue in general dimension, or even the limit $$N\rightarrow\infty$$, i.e. distribution or bounds for the largest eigenvalue. I looked through literatures and didn't find any exact answers. Is there any development on this topic? Or is it still an open problem?

It sounds like you are asking for the distribution of the eigenvalues of a random symmetric matrix with i.i.d. centered entries. This is a huge subject, and you should read the oeuvre of Mehta, or Terry Tao's blog posts, or Anderson/Guionnet/Zeitouni, or..

Mehta, Madan Lal, Random matrices., Pure and Applied Mathematics (Amsterdam) 142. Amsterdam: Elsevier (ISBN 0-12-088409-7/hbk). xviii, 688 p. (2004). ZBL1107.15019.

Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer, An introduction to random matrices, Cambridge Studies in Advanced Mathematics 118. Cambridge: Cambridge University Press (ISBN 978-0-521-19452-5/hbk). xiv, 492 p. (2010). ZBL1184.15023.

ADDENDUM For random correlation matrices, it is a little less clear how you produce them, and the results depend on the model. A detailed discussion is given in:

Holmes, R. B., On random correlation matrices, SIAM J. Matrix Anal. Appl. 12, No. 2, 239-272 (1991). ZBL0723.15020.

(see section 3.2 and the sequel).

• Thanks. I agree that RMT is most likely to answer this. However the off diagonal entries are actually not i.i.d. as the requirements for a matrix to be a "correlation matrix" is 1) diagonal entries are 1; 2) symmetric; 3) semi-positive definite. I believe the complexity is hiding behind the third requiement. – Jiyuan Zhang Nov 6 '18 at 4:16
• @erachang See the addendum. – Igor Rivin Nov 6 '18 at 4:41